iPhone 6 Long screw damage

Unfortunately, it sounds like you have Long-Screw Damage. If you want to be certain before sending your logic board out for repair, I would start by testing a bare-bones setup. Start by disconnecting the battery first (always!). Then disconnect the digitizer, LCD, front camera, rear camera and home button extension flex (see this guide). Now re-connect the battery; you should only have the battery and Lightning Dock flex connected. Connect a known-good, preferably Apple-original Lightning cable to an iTunes enabled computer.

If you have access to a multimeter, you could measure the battery first. If it reads ~3.8V - 4.2V, then it is good. Does iTunes recognize your phone? If so, then you should re-connect one flex at a time until the phone stops working. I would start with the Home button flex first, then add the rear camera, LCD & Digitizer then from camera flex.

If the bare-bones setup does not get recognized by iTunes, then you most likely have a logic board issue. You could send it (or find a local) to a repair shop that specializes in micro-soldering repair. Many shops offer mail-in service and offer a no fix / no fee policy.